University of California, Los AngelesDepartment of StatisticsStatistics C183/C283Instructor: Nicolas ChristouExam 107 May 2010Name:Problem 1(20 points)The betas for 10 stocks in two historical periods 2000-2004 and 2005-2009 are as follows:beta1beta2[1,] 0.9072828 0.7333601[2,] 1.0874136 1.0096048[3,] 0.9871119 1.1143148[4,] 1.0084073 1.1011334[5,] 0.7606293 0.7711888[6,] 0.8047901 0.7834646[7,] 0.9533157 0.9914738[8,] 0.8036708 1.1083840[9,] 1.0867607 0.9524100[10,] 1.0315184 0.8759303a. Explain how you can obtain an estimate for the beta of stock 8 for the period 2010-2014 using theBlume’s technique.b. Suppose that for the second period 2005-2009 the variance of the return of theS&P500index isσ2m= 0.00217. Assume that the single index model holds. Find the covariance between stocks 1 and3 during the same period.c. Explain how you can obtain an estimate for the beta of stock 8 for the period 2010-2014 using theVacicek’s technique.d. Suppose that the correlation coefficient between stockAandS&P500during the period 2005-2009 is0.20. The variance of the return of stockAduring the same period is 0.0143 and the variance of thereturn of theS&P500index wasσ2m= 0.00217. Find the beta of stockA.

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Problem 2(20 points)Use the following for questions (a) and (b) below:Stock¯RσA0.120.20B???0.08It is also given thatρAB= 0.1.a. What expected return on stockBwould result in an optimum portfolio of12Aand12B? Assume shortsales are allowed and thatRf= 0.04.b. What expected return on stockBwould mean that stockBwould not be held? Assume short salesare allowed and thatRf= 0.04.c. SupposeXandYrepresent the returns of two stocks. Show that these two random variablesXandYcannot possibly have the following properties:E(X) = 0.3, E(Y) = 0.2, E(X2) = 0.1, E(Y2) = 0.29,andE(XY) = 0. Reminder:σXY=E(X-μX)(Y-μY) =EXY-(EX)(EY).

Problem 3(20 points)Using the packagestockPortfoliowe have obtained the returns of IBM, Exxon-Mobil, and the S&P500index for the period2005-01-31to2009-12-31. The summary statistics, variance-covariance matrix of the

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